Octal calculator

How the octal calculator works

This online octal calculator can perform addition, subtraction, multiplication, and division on octal numbers. It accepts two or more octal values, executes the chosen mathematical operation, and instantly displays the result in octal form.

The calculation follows three main steps:

1. Convert each octal number to its decimal equivalent.
2. Perform the arithmetic operation in decimal form.
3. Convert the final decimal result back to octal notation.

This approach minimizes manual errors and allows your operations to be performed quickly and accurately.

If you need to convert numbers to octal number system, then use octal converter.

What is the octal numeral system?

The octal numeral system, also known as base-8, is a positional numeral system that uses eight symbols to represent numbers: 0, 1, 2, 3, 4, 5, 6, and 7. Unlike the commonly used decimal system, which employs powers of 10, the octal system uses powers of 8. Each digit in an octal number has a place value that is a power of 8 based on its position from the right.

For example, the octal number 725₈ corresponds to the decimal number:

$7 \times 8^2 + 2 \times 8^1 + 5 \times 8^0 = 448 + 16 + 5 = 469_{10}$

Octal numbers are especially important in computing and digital systems because each octal digit represents exactly three binary digits (bits), making conversions between binary and octal particularly efficient and convenient.

Step-by-step conversion

From octal to decimal

To convert an octal number to decimal, use the following formula:

$N_{10} = d_n \times 8^n + d_{n-1} \times 8^{n-1} + \dots + d_1 \times 8^1 + d_0 \times 8^0$

Where $d_i$ represents each digit in the octal number, starting from the rightmost digit $d_0$.

From decimal to octal

To convert a decimal number to octal, divide the number by 8 repeatedly. Record the remainders from each division. When the quotient becomes zero, reverse the order of the remainders—they form the octal representation.

Examples

Example 1: Octal addition

Compute $27_8 + 13_8 + 4_8$.

Step 1: Convert to decimal: $27₈ = 2*8^1 + 7*8^0 = 23₁₀$ $13₈ = 1*8^1 + 3*8^0 = 11₁₀$ $4₈ = 4*8^0 = 4₁₀$

Step 2: Perform addition: $23₁₀ + 11₁₀ + 4₁₀ = 38₁₀$

Step 3: Convert result to octal:

So the octal result is $46_8$.

Example 2: Octal subtraction

Compute $132_8 - 57_8$.

Step 1: Convert to decimal: $132₈ = 1*64 + 3*8 + 2*8^0 = 90₁₀$ $57₈ = 5*8^1 + 7*8^0 = 47₁₀$

Step 2: Subtraction: $90₁₀ - 47₁₀ = 43₁₀$

Step 3: Convert to octal:

So the octal result is $53_8$.

So $132_8 - 57_8 = 53_8$.

Example 3: Octal multiplication

Compute $23_8 \times 7_8$.

Step 1: Convert to decimal: $23₈ = 2*8^1 + 3*8^0 = 19₁₀$ $7₈ = 7*8^0 = 7₁₀$ $19₁₀ * 7₁₀ = 133₁₀$

Step 2: Convert 133₁₀ into octal:

So $23_8 \times 7_8 = 205_8$.

Example 4: Octal division

Compute $726_8 \div 12_8$.

Step 1: Convert to decimal: $726₈ = 7*64 + 2*8 + 6*8^0 = 470₁₀$ $12₈ = 1*8 + 2*8^0 = 10₁₀$
$470₁₀ ÷ 10₁₀ = 47₁₀$

Step 2: Convert integer part (47) to octal:

So $726_8 \div 12_8 = 57_8$.

Frequently Asked Questions

How to add 63₈ and 45₈?

Convert them first: $63₈ = 6*8^1 + 3*8^0 = 51₁₀$ $45₈ = 4*8^1 + 5*8^0 = 37₁₀$ $51₁₀ + 37₁₀ = 88₁₀$

Convert to octal:

So $63_8 + 45_8 = 130_8$.

What is the octal result of multiplying 56₈ by 3₈?

Convert to decimal: $56₈ = 5*8^1 + 6*8^0 = 46₁₀$ $3₈ = 3*8^0 = 3₁₀$ $46₁₀ * 3₁₀ = 138₁₀$

Convert to octal:

So $212_8$.